Homotopy Colimits Research Articles

Closed categories, lax limits and homotopy limits

Closed categories, lax limits and homotopy limits

Note on sequences of Mayer-Vietoris type

In this largely expository note, we reexamine the construction of the homotopical Mayer-Vietoris sequence associated to a homotopy pullback. We show that in this situation, the Mayer-Vietoris sequence may be realized simply as the homotopy sequence of a suitable fibration. The usual approaches to constructing the Mayer-Vietoris sequence involve some auxiliary algebraic result, such as the Barratt-Whitehead lemma; the present approach avoids any such considerations. An additional beneficial feature of our approach is the attention paid to the bottom end of the Mayer-Vietoris sequence. Thus we are led to a cleaner proof of Proposition II.7.11 of [HMR]; moreover, we show that the converse of this latter result is also true. The homological Mayer-Vietoris sequence associated to a homotopy pushout may be established in a very similar manner, as we point out at the end of the paper.

Homotopy Pullbacks and the Hopf Invariant

Journal of the London Mathematical SocietyVolume s2-19, Issue 1 p. 153-158 Notes and Papers Homotopy Pullbacks and the Hopf Invariant Marshall Walker, Marshall Walker York University, Downsview, OntarioSearch for more papers by this author Marshall Walker, Marshall Walker York University, Downsview, OntarioSearch for more papers by this author First published: February 1979 https://doi.org/10.1112/jlms/s2-19.1.153Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-19, Issue1February 1979Pages 153-158 RelatedInformation

Homotopy colimits in the category of small categories

In (13), Quillen defines a higher algebraic K-theory by taking homotopy groups of the classifying spaces of certain categories. Certain questions in K-theory then become questions such as when do functors induce a homotopy equivalence of classifying spaces, or when is a square of categories homotopy cartesian? Quillen has given some techniques for answering such questions. F. Waldhausen has extended these ideas in (19), and broadened the range of applications to include geometric topology (20).

Homotopy Pull-Backs and Applications to Duality

The topic of homotopy pull-backs and push-outs has recently been discussed by a number of authors; Boardman and Vogt [5], Bousfield and Kan [6], Fantham [7], Mather [11], and Vogt [16]. Mather develops the theory with an eye to applications and of particular interest is his cube theorem which appears in this paper as Theorem (1.10); the significance of this theorem to applications is shown in [11]. As often occurs in homotopy theory the dual is not true. The purpose of this paper is to examine approximations to the dual in order to obtain new information concerning classical problems of duality.

Correction to "Shapes of Complexes, Ends of Manifolds, Homotopy Limits and the Wall Obstruction"

In [1], Theorem 1.1 ((iii) (ii)) and 3.3, we claimed that shape dominated by a complex implies shape equivalent to a complex, in both unpointed and pointed shape theory. There is an error in our proof, which is easily corrected im the pointed (connected) case as follows: In Theorem 2.1 assume the additional hypothesis that each Ye, is connected; in the proof of 2.1 we cannot conclude from the vanishing of the kernel that P. is injective (since exactness is of pointed sets not groups); however Y is connected (read the exact sequence with L the 0-sphere: the kernel vanishes and p# is onto a point); and p,: wi( Y) lim {~r( Yj)} is an isomorphism for all i (exactness for groups); thus po(d'ou)= 1 as before, and (d'ou)op: Ye Yis a weak homotopy equivalence, hence a homotopy equivalence, hence the identity. The required implication (iii) (ii) of Theorem 1.1 follows easily in the pointed connected case. (We have included the details in [2].) The unpointed case is open, since in 2.1 we are now assuming all Yn's connected. We are able to prove that shape equivalent to a complex implies pointed shape equivalent to a complex, and that shape dominated by a complex implies shape equivalent to a complex in the unpointed (connected) case if liml of the fundamental groups vanishes; but the latter hypothesis may be redundant. Note that the rest of the paper is unaffected by this correction, except that, in 7.4, Z should be connected, and the domination pointed. We thank J. Dydak for pointing out the error.

Homotopy Limits and the Homotopy Type of Functor Categories

Let ${\mathbf {Y}}:I \to$ Simplicial Sets be a functor. We give a sufficient condition for the map ${\text {ho}}\lim \limits _ \to {\mathbf {Y}} \to \lim \limits _ \to {\mathbf {Y}}$ to be a weak equivalence. Then we apply this to determine the Artin-Mazur homotopy type of the functor category Funct($I$, Sets).

Pull-Backs in Homotopy Theory

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no essential map between Eilenberg-MacLane spaces of different dimensions has a kernel. In this paper we define homotopy pull-backs and push-outs, which do exist and which behave like pull-backs and push-outs, and we give some of their properties. Applications may be found in [3; 5; 6 and 14].

Homotopy limits and the homotopy type of functor categories

Let ${\mathbf {Y}}:I \to$ Simplicial Sets be a functor. We give a sufficient condition for the map ${\text {ho}}\lim \limits _ \to {\mathbf {Y}} \to \lim \limits _ \to {\mathbf {Y}}$ to be a weak equivalence. Then we apply this to determine the Artin-Mazur homotopy type of the functor category Funct($I$, Sets).

Shapes of Complexes, Ends of Manifolds, Homotopy Limits and the Wall Obstruction

Shapes of Complexes, Ends of Manifolds, Homotopy Limits and the Wall Obstruction

A remark on ?homotofp fibrations?

We define a category TopF of “homotopy fibrations with fibre F” (or rather “maps with homotopy fibre F”) and show that this category is closed under certain colimits and “homotopy colimits”. It follows that the geometric realization of a semisimplicial object in TopF is again in TopF. As a corollary we show that for a homotopy everything H-space A*(i.e. a (special) Γ-space in the sense of G. Segal (s.[9],[10])) with homotopy inverse the loop space of the classifying space of A* is homotopy equivalent (not only weakly s. [9],[iO])to A1 even without assuming that all spaces involved have the homotopy type of CW-complexes (compare [8]).

Homotopy limits and colimits

Homotopy limits and colimits

Homotopy colimits of diagrams over posets and variations on a theorem of Thomason

We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen's Theorem A for posets.

Cech approximation to the Brown-Gersten spectral sequence

In this paper, we show that the etale index of a torsion cohomological Brauer class is divisible by the period of the class. The tool used to make this computation is the Cech approximation of the title. To create the approximation, we use the folklore theorem that the homotopy limit and Postnikov spectral sequences for a cosimplicial space agree beginning with the E2-page. As far we know, this folklore theorem has no proof in the literature, so we include a proof.

Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions

Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting these (co)limits and that they are preserved by the functors in the limit cone. In particular, the Bousfield-Kan homotopy limit of a diagram of quasi-categories admit any limits or colimits existing in and preserved by the functors in that diagram. In previous work, we demonstrated that the quasi-category of algebras for a homotopy coherent monad could be described as a weighted limit with projective cofibrant weight, so these results immediately provide us with important (co)completeness results for quasi-categories of algebras. These generalise most of the classical categorical results, except for a well known theorem which shows that limits lift to the category of algebras for any monad, regardless of whether its functor part preserves those limits. The second half of this paper establishes this more general result in the quasi-categorical setting: showing that the monadic forgetful functor of the quasi-category of algebras for a homotopy coherent monad creates all limits that exist in the base quasi-category, without further assumption on the monad. This proof relies upon a more delicate and explicit analysis of the particular weight used to define quasi-categories of algebras.

Rigidity of the $K(1)$-local stable homotopy category

One goal of algebraic topology is to find algebraic invariants that classify topological spaces up to homotopy equivalence. The notion of homotopy is not only restricted to topology. It also appears in algebra, for example as a chain homotopy between two maps of chain complexes. The theory of model categories,introduced by D. Quillen [Qui06], provided us with a powerful common language to represent different notions of homotopy. Quillen’s work transformed algebraic topology from the study of topological spaces into a wider setting useful in many areas of mathematics, such as homological algebra and algebraic geometry, where homotopy theoretic approaches led to interesting results. In brief, a model structure on a category C is a choice of three distinguished classes of morphisms: weak equivalences, fibrations, and cofibrations, satisfying certain axioms. We can pass to the homotopy category Ho(C) associated to a model category C by inverting the weak equivalences, i.e. by making them into isomorphisms. While the axioms allow us to define the homotopy relations between classes of morphisms in C, the classes of fibrations and cofibrations provide us with a solution to the set-theoretic issues arising in general localisations of categories. Even though it is sometimes sufficient to work in the homotopy category, looking at the homotopy level alone does not provide us with enough higher order structure information. For example, homotopy (co)limits are not usually a homotopy invariant, and in order to define them we need the tools provided by the model category. This is where the question of rigidity may be asked: if we just had the structure of the homotopy category, how much of the underlying model structure can we recover? This question of rigidity has been investigated during the last decade, and an extremely small list of examples have been studied, which leaves us with a lot of open questions regarding this fascinating subject. Ou goal is to investigate one of the open questions which have not been answered before. In this thesis, we prove rigidity of the K(1)-local stable homotopy category Ho(LK(1)Sp) at p = 2. In other words, we show that recovering higher order structure information, which is meant to be lost on the homotopy level,is possible by just looking at the triangulated structure of Ho(LK(1)Sp). This new result does not only add one more example to the list of known examples of rigidity in stable homotopy theory, it is also the first studied case of rigidity in the world of Morava K-theory.

Euler characteristics of finite homotopy colimits

In this short note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by a finite category, but the behavior is more uniform when we index by finite quasicategories instead. The formula simultaneously generalizes the additive formula for Euler characteristic of a homotopy pushout and the multiplicative formula for Euler characteristic of a fiber bundle.

On left and right model categories and left and right Bousfield localizations

We verify the existence of left Bouseld localizations and of enriched left Bouseld localizations, and we prove a collection of useful technical results characterizing certain brations of (enriched) left Bouseld localizations. We also use such Bouseld localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition. We then verify the existence of right Bouseld localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.